Preheat grill or skillet to medium-high heat.
Brush chicken with 1 tablespoon olive oil and season with 1/4 teaspoon each salt and pepper.
Grill chicken, uncovered, until internal temperature reaches 165°F, about 5 to 6 minutes per side. Let stand 5 minutes, then cut into slices.
Meanwhile, whisk together 2 tablespoons olive oil, white balsamic vinegar, honey, mustard, 1/2 teaspoon salt, and 1/4 teaspoon pepper in a large bowl. Add arugula, basil, and onion; toss to coat.
In a small bowl or shallow dish, mash together blackberries with a splash of white balsamic vinegar, using a fork.
Spread blackberry mixture onto one side of each Sandwich Potato Roll OR half of the toasted Butter Bread slices.
Layer sandwiches with salad mixture, sliced grilled chicken, and crumbled goat cheese. Top with roll OR second slice of bread. Serve.